Tuesday, January 4, 2011

Mathemagic

Arthur Benjamin does "Mathemagic"

In a lively show, mathemagician Arthur Benjamin races a team of calculators to figure out 3-digit squares, solves another massive mental equation and guesses a few birthdays. How does he do it? He’ll tell you.

TranscriptWell, good morning ladies and gentlemen. My name is Art Benjamin, and I am a "mathemagician." What that means is, I combine my loves of math and magic to do something I call "mathemagics." But before I get started, I have a quick question for the audience. By any chance, did anyone happen to bring with them this morning a calculator? Seriously, if you have a calculator with you, raise your hand, raise your hand.

I -- was -- your hand go up? Now bring it out, bring it out. Anybody else? I see, I see one way in the back. You sir, that's three, And anybody on this side here? OK, you over there on the aisle. Would the four of you with calculators please bring out your calculators, then join me up on stage. And let's give these volunteers a nice round of applause.

(Applause)

That's right. Now, since I haven't had the chance to work with these calculators, I need to make sure that they are all working properly. Would somebody get us started by giving us a two-digit number please? How about a two-digit number?

Audience: 22.

Arthur Benjamin: 22. And another two-digit number, Sir?

Audience: 47.

AB: Multiply 22 times 47, make sure you get 1,034, or the calculators are not working. Do all of you get 1,034? 1,034?

Woman: No.

AB: 594. Let's give three of them a nice round of applause there.

(Applause)

Would you like to try a more standard calculator, just in case? OK, great. What I'm going to try and do then -- I notice that took some of you a little bit of time to get your answer. That's OK. I'll give you a shortcut for multiplying even faster on the calculator. There is something called the square of a number, which most of you know is taking a number and multiplying it by itself. For instance, five squared would be?

Audience: 25.

AB: 25. Now, the way we can square on most calculators -- let me demonstrate with this one -- is by taking the number, such as five, hitting "times" and then "equals," and on most calculators that will give you the square. On some of these ancient RPN calculators, you've got an "x squared" button on it, will allow you to do the calculation even faster. What I'm going to try and do now is to square, in my head, four two-digit numbers faster than they can do on their calculators, even using the shortcut method. What I'll use is the second row this time, and I'll get four of you -- one, two, three, four -- to each yell out a two-digit number, and if you would square the first number, and if you would square the second, the third and the fourth, I will try and race you to the answer. OK? So quickly -- a two-digit number please.

Audience: 37.

AB: 37 squared, OK.

Audience: 23.

AB: 23 squared, OK.

Audience: 59.

AB: 59 squared, OK, and finally?

Audience: 93.

AB: 93 squared. Would you call out your answers, please?

Woman: 1369. AB: 1369.

Woman: 529.

AB: 529.

Man: 3481.

AB: 3481.

Man: 8649.

AB: Thank you very much.

(Applause)

Let me try to take this one step further. I'm going to try to square some three-digit numbers this time. I won't even write these down -- I'll just call them out as they're called out to me. Anyone I point to, call out a three-digit number. Anyone on our panel, verify the answer. Just give some indication if it's right. A three-digit number, sir, yes?

I'm going to try to square a four-digit number this time. Now you can all take your time on this; I will not beat you to the answer on this one, but I will try to get the answer right. To make this a little bit more random, let's take the fourth row this time, let's say, one, two, three, four. If each of you would call out a single digit between zero and nine, that will be the four-digit number that I'll square.

Audience: Nine.

AB: Nine.

Audience: Seven. AB: Seven.

Audience: Five. AB: Five.

Audience: Eight. AB: Eight. 9,758, this will take me a little bit of time, so bear with me. 95,218,564?

Woman: Yes.

AB: Thank you very much.

(Applause)

Now, I would attempt to square a five-digit number -- and I can -- but unfortunately most calculators cannot.

AB: I'll talk to you later. In the meanwhile, let me conclude the first part of my show by doing something a little trickier. Let's take the largest number on the board here, 8649. Would you each enter that on your calculator? And instead of squaring it this time, I want you to take that number and multiply it by any three-digit number that you want, but don't tell me what you're multiplying by -- just multiply it by any random three-digit number. So you should have as an answer either a six-digit or probably a seven-digit number. How many digits do you have, six or seven?

Seven, and yours? Woman: Seven.

AB: Seven? Seven? And, uncertain.

Man: Yeah.

AB: Seven. Is there any possible way that I could know what seven digit numbers you have? Say "No."

(Laughter)

Good. Then I shall attempt the impossible -- or at least the improbable. What I'd like each of you to do is to call out for me any six of your seven digits, any six of them, in any order you'd like.

(Laughter)

One digit at a time, I shall try and determine the digit you've left out. So, starting with your seven-digit number, call out any six of them please.

Woman: One, OK, 197042.

AB: Did you leave out the number 6?

Woman: Yes, AB: Good, OK, that's one. You have a seven-digit number, call out any six of them please.

Woman: 44875.

AB: I think I only heard five numbers. I -- wait -- 44875 -- did you leave out the number 6?

Woman: Yes. AB: Same as she did, OK. You've got a seven-digit number -- call out any six of them loud and clear.

Man: 079044.

AB: I think you left out the number 3? That's three. The odds of me getting all four of these right by random guessing would be one in 10,000: 10 to the fourth power. OK, any six of them. Really scramble them up this time, please.

Man: 263972.

AB: Did you leave out the number 7? And let's give all four of these people a nice round of applause. Thank you very much. For my next number -- (Laughter) while I mentally recharge my batteries, I have one more question for the audience. By any chance, does anybody here happen to know the day of the week that they were born on? If you think you know your birth day, raise your hand. Let's see, starting with -- let's start with a gentleman first, OK sir, what year was it, first of all? That's why I start with a gentleman first. What year?

Audience: 1953.

AB: 1953, and the month?

Audience: November. AB: November what?

Audience: 23rd.

AB: 23rd -- was that a Monday? Audience: Yes. Yes, good. somebody else? Who else would like -- see I don't -- haven't seen any women's hands up. OK, it's -- how about you, what year?

Audience: 1949. AB: 1949, and the month?

Audience: October. AB: October what?

Audience: Fifth.

AB: Fifth -- was that a Wednesday? Yes, my -- I'll go way to the back right now, how about you? Yell it out, what year? Audience: 1959.

AB: 1959, OK -- and the month?

Audience: February.

AB: February what? Audience: Sixth.

AB: Sixth -- was that a Friday? Audience: Yes. Good, how about the person behind her? Call -- call -- what year was it?

Audience: 1947. AB: 1947, and the month?

Audience: May. AB: May what?

Audience: Seventh. AB: Seventh -- would that be a Wednesday?

Audience: Yes.

AB: Thank you very much.

(Applause)

Anybody here who'd like to know the day of the week they were born? We can do it that way. Of course, I could just make up an answer and you wouldn't know, so I come prepared for that. I brought with me a book of calendars. It goes as far back into the past as 1800, 'cause you never know.

(Laughter)

I didn't mean to look at you, sir -- you were just sitting there. Anyway, Chris, you can help me out here, if you wouldn't mind. This is a book of calendars, and I'll ask -- who was it that wanted to know their birth day? You sir? OK. What year was it, first of all?

Audience: 1966.

AB: '66 -- turn to the calendar with 1966 -- and what month?

Audience: April. AB: April what?

Audience: 17th. AB: 17th -- I believe that was a Sunday. Can you confirm, Chris?

Chris Anderson: Yes. AB: Yeah, OK. I'll tell you what, Chris: as long as you have that book in front of you, do me a favor, turn to a year outside of the 1900s, either into the 1800s or way into the 2000s -- that'll be a much greater challenge for me. What year, Chris, would you like?

CA: 1824.

AB: 1824, OK. And what month? CA: June.

AB: June what? CA: Sixth.

AB: Sixth -- was that a Sunday?

CA: It was. AB: And it was cloudy. Good, thank you very much.

(Applause)

But I'd like to wrap things up now by alluding to something from earlier in the presentation. There was a gentleman up here who had a 10-digit calculator. Where is he, would you stand up, 10-digit guy? OK, well stand up for me just for a second, so I can see where you are. OK, oh, OK -- you have a 10-digit calculator, sir, as well? OK, what I'm going to try and do, is to square in my head a five-digit number requiring a 10-digit calculator. But to make my job more interesting for you, as well as for me, I'm going to do this problem thinking out loud. So you can actually, honestly hear what's going on in my mind while I do a calculation of this size.

Now, I have to apologize to our magician friend Lennart Green. I know as a magician we're not supposed to reveal our secrets, but I'm not too afraid that people are going to start doing my show next week, so -- I think we're OK. So, let's see, let's take a -- let's take a different row of people, starting with you. I'll get five digits: one, two, three, four -- oh, I did this row already. Let's do the row before you, starting with you sir: one, two, three, four, five. Call out a single digit -- that will be the five-digit number that I will try to square. Go ahead.

Audience: Five. AB: Five.

Audience: Seven. AB: Seven.

Audience: Six. AB: Six.

Audience: Eight. AB: Eight.

Audience: Three. AB: Three. 57,683 squared. Yuck.

Let me explain to you how I'm going to attempt this problem. I'm going to break the problem down into three parts. I'll do 57,000 squared, plus 683 squared, plus 57,000 times 683 times two. Add all those numbers together, and with any luck, arrive at the answer. Now, let me recap. Thank you. While I explain something else -- -- I know, that you can use, right? While I do these calculation, you might hear certain words, as opposed to numbers, creep into the calculation. Let me explain what that is. This is a phonetic code, a mnemonic device that I use, that allows me to convert numbers into words. I store them as words, and later on retrieve them as numbers. I know it sounds complicated; it's not -- I just don't want you to think you're seeing something out of "Rain Man" here.

If you want to talk to me about ADHD afterwards, you can talk to me then. All right -- by the way, one last instruction, for my judges with the calculators -- OK, you know who you are -- there is at least a 50 percent chance that I will make a mistake here. If I do, don't tell me what the mistake is; just say, "you're close," or something like that, and I'll try and figure out the answer -- which could be pretty entertaining in itself. If, however, I am right, whatever you do, don't keep it to yourself, OK?

(Laughter)

Make sure everybody knows that I got the answer right, because this is my big finish, OK. So, without any more stalling, here we go. I'll start the problem in the middle, with 57 times 683. Now, 57 times 68 is 3,400, plus 476 is 3876, that's 38,760 plus 171, 38,760 plus 171 is 38,931. 38,931; double that to get 77,862. 77,862 becomes cookie fission, cookie fission is 77,822. That seems right, I'll go on. Cookie fission, OK. Next, I do 57 squared, which is 3,249, so I can say, three billion. Take the 249, add that to cookie, 249, oops, but I see a carry coming -- 249 -- add that to cookie, 250 plus 77, is 327 million -- fission, fission, OK, finally, we do 683 squared, that's 700 times 666, plus 17 squared is 466,489, rev up if I need it, rev up, take the 466, add that to fission, to get, oh gee -- 328,489.